# Marginal conditional stochastic dominance

In finance, marginal conditional stochastic dominance is a condition under which a portfolio can be improved in the eyes of all risk-averse investors by incrementally moving funds out of one asset (or one sub-group of the portfolio's assets) and into another.[1][2][3] Each risk-averse investor is assumed to maximize the expected value of an increasing, concave von Neumann-Morgenstern utility function. All such investors prefer portfolio B over portfolio A if the portfolio return of B is second-order stochastically dominant over that of A; roughly speaking this means that the density function of A's return can be formed from that of B's return by pushing some of the probability mass of B's return to the left (which is disliked by all increasing utility functions) and then spreading out some of the density mass (which is disliked by all concave utility functions).

If a portfolio A is marginally conditionally stochastically dominated by some incrementally different portfolio B, then it is said to be inefficient in the sense that it is not the optimal portfolio for anyone. Note that this context of portfolio optimization is not limited to situations in which mean-variance analysis applies.

The presence of marginal conditional stochastic dominance is sufficient, but not necessary, for a portfolio to be inefficient. This is because marginal conditional stochastic dominance only considers incremental portfolio changes involving two sub-groups of assets — one whose holdings are decreased and one whose holdings are increased. It is possible for an inefficient portfolio to not be second-order stochastically dominated by any such one-for-one shift of funds, and yet to by dominated by a shift of funds involving three or more sub-groups of assets.[4]

## Testing

Yitzhaki and Mayshar[5] presented a linear programming-based approach to testing for portfolio inefficiency which works even when the necessary conditional of marginal conditional stochastic dominance is not met. Other similar tests have also been developed.[6][7][8][9]

## References

1. ^ Shalit, H., and Yitzhaki, S. "Marginal conditional stochastic dominance," Management Science 40, 1994, 670-684.
2. ^ Chow, K. V., "Marginal stochastic dominance, statistical inference, and measuring portfolio performance," Journal of Financial Research 24, 2001, 289-307.
3. ^ Post, T., "On the dual test for SSD efficiency: with an application to momentum investment strategies," European Journal of Operational Research 185(3), 2008, 1564-1573.
4. ^ Zhang, Duo, "A demonstration of the non-necessity of marginal conditional stochastic dominance for portfolio inefficiency," The Quarterly Review of Economics and Finance 49, May 2009, 417-423.
5. ^ Yitzhaki,Shlomo, and Mayshar, Joram. "Characterizing Efficient Portfolios", SSRN Working Paper, [1]
6. ^ Post, T., "Empirical tests for stochastic dominance efficiency," Journal of Finance 58(5), 2003, 1905-1932.
7. ^ Kuosmanen, T., "Efficient diversification according to stochastic dominance criteria," Management Science 50, 2004, 1390-1406.
8. ^ Post, T., and Levy, H., "Does risk seeking drive stock prices? A stochastic dominance analysis of aggregate investor preferences and beliefs," Review of Financial Studies 18, 2005, 925-953.
9. ^ Post, T., and Versijp, P., "Multivariate tests for stochastic dominance efficiency of a given portfolio," Journal of Financial and Quantitative Analysis 42(2), 2007, 489-516.